The coins the Illusionist gives the poor children each have a ring surrounding a differently colored disc. The technology to make those coins is only a few decades old. They're Czech 50 heller coins, introduced in 1993.
As Eisenheim puts stacks of notes into his leather briefcase at the bank, a silver zipper is visible on the bag. Some early prototype of zippers did exist, but were not in common use well into the 20th century.
When Inspector Uhl finds Leopold to tell him of Sophie and Eisenheim's relationship, Leopold asks "What where they doing? Touching? Kissing? Fornicating?" His face is only shown when he says "fornicating", but he mouths a completely different word. (see Trivia)
When Uhl opens The Orange Tree folder in the theatre and sees the plans for the locket for the first time, there is a hole in the center of the sheet of paper. When Uhl is in the stables and looks at the plans a second time, the hole has disappeared.
At around the 20:17 mark, When Inspector Uhl is inspecting the props backstage, Eisenheim has his arms at his side during their conversation. Then suddenly he is facing the inspector at a different angle and his arms are crossed.
Duchess Sophie von Teschen reveals to Eduard that Crown Prince Leopold intends to win over the Hungarians by aligning himself with her family. In reality, the Duchy of Teschen was part of the Austrian half of the Empire, so even if the Crown Prince had married her, it would not have had that much of a impact on the Empire's Hungarian citizens.
Crown Prince Leopold (Rufus Sewell) wears a mustache that is clearly fake. However, a real crown prince of that era would have no reason to wear a fake mustache and this is never explained, as the audience is presumably expected to believe it is the real deal.
The Crown Prince has supposedly murdered a former girlfriend, to hide the fact that he had been beating her with a whip. But in that era, men regularly did whip their wives or mistresses. There was even some social approval. Whereas murder obviously was unacceptable.
When Inspector Uhl is investigating Eisenheim's latest illusion of summoning spirits, one of his aides shows an early turn-of-the-century movie projector portraying a color/sepia-based image of a person. Turn-of-the-century film was often hand-colorized using stencil methods. Georges Méliès's fantasy shorts used that method. Several genuine color film technologies were also in use by the early 1920's.
The blood from Sophie was on the right side of the horse yet her wound is on the left side. But, as she rides the horse out of the stables, she is slumped over the right side, her left side pressing against the right side of the horse.
When giving a private exhibition to the Crown Prince and assembled nobles, Eisenheim explains the legend of King Arthur and the Sword in the Stone, referring to the sword as Excalibur. Though popular culture does commonly refer to the sword as such, Excalibur was actually a different sword, given to Arthur by the Lady of the Lake after the sword he pulled from the stone was broken. This is one version - Arthurian legend is hardly 100% factual history, and many incarnations of Arthur indeed conflate the two swords - Excalbur and Claret.
The goof items below may give away important plot points.
When the chief inspector (Giamatti) finds the locket and gemstone in the stable, it is after finding the locket diagram, as he consults it in the stable. This means that it is after Eisenheim has disappeared, not a flashback to an original examination of the crime scene. The stable in a royal crown prince's hunting lodge would have enough activity (mucking and resetting bedding) for that to have been found or discarded especially by a crown prince trying to cover up a crime. While the time period is indeterminate it does give the impression of at least weeks if not months or a year. Remember that it could not be recently planted evidence because in the original if hasty examination of the crime scene the glint is shown, obvious in retrospect to be the gemstone.
After the Crown Prince Leopold commits suicide, he is breathing, as his stomach moves up and down. Even when someone commits suicide by shooting himself in the head, the heart will usually beat for a few more seconds since it is controlled largely by a nervous system internal to the heart. The person is also usually able to take a few more breaths before the entire body shuts down completely as long as the breathing center of the brain is undamaged.
After Princess Sophie is "murdered", several weeks apparently pass while Eisenheim prepares his new show, yet inspector Uhl returns to the stable and finds the gemstone and the locket after Eisenheim's disappearance. After all those weeks the stall was never cleaned. The stall did not appear to be secured in any manner generally used for a crime scene.